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searching for Differentiable function 167 found (298 total)

alternate case: differentiable function

Piecewise function (1,065 words) [view diff] no match in snippet view article find links to article

In mathematics, a piecewise function (also called a piecewise-defined function, a hybrid function, or a function defined by cases) is a function whose

derivative is equal to the right derivative. An example of a semi-differentiable function, which is not differentiable, is the absolute value function f

its local minima, but need not actually be convex. Informally, a differentiable function is pseudoconvex if it is increasing in any direction where it has

{x}})} with an equilibrium point at the origin, a continuously differentiable function V(x) such that the origin is a boundary point of the set G = {

implementable algorithm. They are called proximal because each non-differentiable function among f 1 , . . . , f n {\displaystyle f_{1},...,f_{n}} is involved

necessary condition for all values of k {\displaystyle k} . Consider a differentiable function f ( x ) {\displaystyle f(x)} on [ a , b ] {\displaystyle [a,b]}

mathematics, the Fabius function is an example of an infinitely differentiable function that is nowhere analytic, found by Jaap Fabius (1966). This function

stochastic processes, a harmonic function is a twice continuously differentiable function f : U → R , {\displaystyle f\colon U\to \mathbb {R} ,} where U

and real-differentiability. On the real line, for example, the differentiable function f ( x ) = x 2 {\displaystyle f(x)=x^{2}} is not an open map, as

for sparse or partially separable problems. A twice continuously differentiable function x ↦ f ( x ) {\displaystyle x\mapsto f(x)} has a gradient ( ∇ f

{\displaystyle u:\Omega \rightarrow \mathbb {R} } be a weakly differentiable function that vanishes on the boundary of Ω {\displaystyle \Omega } in the

transposition, f(e) is scalar function, and f(0) = 0. Suppose, f(e) is a differentiable function and the following condition k 1 < f ′ ( e ) < k 2 . {\displaystyle

point of the closed set if there is a real-valued continuously differentiable function maximized locally at the point with that vector as its derivative

_{j}\left(U_{i}\cap U_{j}\right)} is an r {\displaystyle r} -times continuously differentiable function for every i , j ∈ I ; {\displaystyle i,j\in I;} that is, the r

series is a Fourier series type expansion of twice continuously differentiable function in the interval ( 0 , π ) {\displaystyle (0,\pi )} in terms of

Fermat's theorem in calculus, stating that at any point where a differentiable function attains a local extremum its derivative is zero. In Lagrangian

it can integrate the approximate derivative of an approximately differentiable function (see below for definitions). To do this, one first finds a condition

approximating solutions to equations. Given a twice continuously differentiable function f {\displaystyle f} of one real variable, Taylor's theorem for

versions of the Wirtinger inequality: Let y be a continuous and differentiable function on the interval [0, L] with average value zero and with y(0) =

case where the search space has underlying structure (e.g., is a differentiable function) that can be exploited more efficiently (e.g., Newton's method

is an intensive quantity, i.e., a single-valued, continuous and differentiable function of three-dimensional space (often called a scalar field), i.e.

differentiable vector-valued function from Euclidean space is a differentiable function valued in a topological vector space (TVS) whose domains is a subset

→ R {\displaystyle f\colon I\to \mathbb {R} } be a real-valued differentiable function. Then f ′ {\displaystyle f'} has the intermediate value property:

distribution of compact support and f {\displaystyle f} is an infinitely differentiable function, the expression v ( f ) = v ( x ↦ f ( x ) ) {\displaystyle v(f)=v(x\mapsto

variable with expectation μ and variance σ2. Further suppose g is a differentiable function for which the two expectations E ⁡ ( g ( X ) ( X − μ ) ) {\displaystyle

non-differentiable functions (an example of an everywhere non-differentiable function is the Weierstrass function). Singh published about a dozen papers

underlying topological space and to each p-times continuously differentiable function the underlying continuous function of topological spaces. Similarly

variational principle. The classical Fermat's theorem says that if a differentiable function attains its minimum at a point, and that point is an interior point

Peter Gustav Lejeune Dirichlet. Given an open set Ω ⊆ Rn and a differentiable function u : Ω → R, the Dirichlet energy of the function u is the real number

\int _{a}^{b}e^{Mf(x)}\,dx,} where f {\displaystyle f} is a twice-differentiable function, M {\displaystyle M} is a large number, and the endpoints a {\displaystyle

10 17 {\displaystyle 10^{15}-10^{17}} eV) by the continuously differentiable function of energy E {\displaystyle E} taking into account the rate of change

whose derivative is nowhere vanishing, i.e. a strictly monotone differentiable function, either increasing or decreasing. The space of such functions consists

D_{q}f(x)={\frac {f(qx)-f(x)}{(q-1)x}}.} For x nonzero, if f is a differentiable function of x then in the limit as q → 1 we obtain the ordinary derivative

pairwise distinct points x0, ..., xn in the domain of an n-times differentiable function f there exists an interior point ξ ∈ ( min { x 0 , … , x n } ,

the inverse function theorem which states that a continuously differentiable function between Euclidean spaces whose derivative matrix is invertible

n → R {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} } be a differentiable function and let v {\displaystyle \mathbf {v} } be a vector in R n {\displaystyle

value of conjunction P ∧ Q {\displaystyle P\land Q} is a twice differentiable function f {\displaystyle f} of truth values of the two propositions P {\displaystyle

X, then it vanishes everywhere in the interior of X. If f is a differentiable function defined on an interval I ⊆ R, then f is logarithmically convex

{\displaystyle u(x)\in C_{c}^{\infty }(\mathbb {R} ^{n})} , infinitely differentiable function with compact support. Introduce u λ ( x ) := u ( λ x ) {\displaystyle

the curve does not vanish. Analytically, r(t) is a three times differentiable function of t with values in R3 and the vectors r ′ ( t ) , r ″ ( t ) {\displaystyle

sequence Illustration of the central limit theorem An infinitely differentiable function that is not analytic Leech lattice Lewy's example on PDEs List

statistical model, and θ {\textstyle \theta } is a continuously differentiable function of φ {\textstyle \varphi } , we say that the prior p θ ( θ ) {\textstyle

Critical point (aka stationary point), any value v in the domain of a differentiable function of any real or complex variable, such that the derivative of v

it is measurable and its Lebesgue measure is zero as well. Any differentiable function has the Luzin N property. This extends to functions that are differentiable

Differentiable vector-valued functions from Euclidean space – Differentiable function in functional analysis Infinite-dimensional vector function – function

that is transformed by continuously differentiable function. If Y = g(X) is a continuous differentiable function, then the density of Y can be written

suppose that the surface is the graph of a twice continuously differentiable function, z = f(x,y), and that the plane z = 0 is tangent to the surface

{1}{2}}f(X_{t})f'(X_{t})\mathrm {d} t+f(X_{t})\mathrm {d} W_{t}} for a given differentiable function f {\displaystyle f} is equivalent to the Stratonovich SDE d X t

integral. It is also possible to show a converse – that every differentiable function is equal to the integral of its derivative, but this requires a

S_{D}:{\vec {r}}={\vec {r}}(u,v),\quad (u,v)\in D} with a continuously differentiable function r → . {\displaystyle {\vec {r}}.} The area of an individual piece

X is an Rm valued semimartingale and f is a twice continuously differentiable function from Rm to Rn, then f(X) is a semimartingale. This is a consequence

x_{0}} . Alternatively, a real analytic function is an infinitely differentiable function such that the Taylor series at any point x 0 {\displaystyle x_{0}}

positive, continuous, nondecreasing function. Then there exists a differentiable function G : [ 0 , ∞ ) → [ 0 , ∞ ) {\displaystyle G:[0,\infty )\rightarrow

\mathrm {d} W_{t},} where f {\displaystyle f} is any continuously differentiable function of two variables W {\displaystyle W} and t {\displaystyle t} and

f ( x ) . {\displaystyle A=Df(x).} However, not every Gateaux differentiable function is Fréchet differentiable. This is analogous to the fact that the

{\displaystyle 0,} making the goal to prove that an everywhere differentiable function whose derivative is always zero must be constant: Choose a real

certain symbol class. For instance, if P(x,ξ) is an infinitely differentiable function on Rn × Rn with the property | ∂ ξ α ∂ x β P ( x , ξ ) | ≤ C α

exists in trace-class norm. If g ( t ) {\displaystyle g(t)} is a differentiable function with values in trace-class operators, then so too is exp ⁡ g (

real-valued function, Du is the gradient of u, and F is a continuously differentiable function that is regular in Du. Suppose that u(x;a) is an m-parameter family

leading up to an example of John Myhill of a computable continuously differentiable function whose derivative is not computable, the remaining two parts of

which states in the deterministic case, that for a continuously differentiable function f ∈ C 1 ( R k , R d ) {\displaystyle f\in C^{1}(\mathbb {R} ^{k}

Itô's lemma. In its simplest form, for any twice continuously differentiable function f on the reals and Itô process X as described above, it states

{\displaystyle U} . By Cauchy's integral formula, which shows that a differentiable function is in fact infinitely differentiable, a function f {\displaystyle

expectation with respect to γn. The gradient operator ∇ acts on a (differentiable) function φ : Rn → R to give a vector field ∇φ : Rn → Rn. The divergence

of coordinates from one coordinate map to another is always a differentiable function. In geometry and kinematics, coordinate systems are used to describe

with 2 input parameters consisting of (1) a real continuously differentiable function on a subinterval of the real numbers and (2) an n {\displaystyle

injective. For example, in calculus if f {\displaystyle f} is a differentiable function defined on some interval, then it is sufficient to show that the

Differentiable vector–valued functions from Euclidean space – Differentiable function in functional analysisPages displaying short descriptions of redirect

bundle. An extremely special case of this is the following: if a differentiable function from reals to the reals has nonzero derivative at a zero of the

way that correctly accounts for a change-of-coordinates. Given a differentiable function f : R n → R m {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R}

function theorem gives a sufficient condition for a continuously differentiable function to be (among other things) locally injective. Every fiber of a

\nu \}} . p-variation Hardy, G. H. (1916). "Weierstrass's Non-Differentiable Function". Transactions of the American Mathematical Society. 17 (3): 301–325

parametrizations of an estimation problem, and θ is a continuously differentiable function of η, then I η ( η ) = I θ ( θ ( η ) ) ( d θ d η ) 2 {\displaystyle

as infinite or zero capital accumulation. Given a continuously differentiable function f : X → Y {\displaystyle f\colon X\to Y} , where X = { x : x ∈

(Statement of the theorem ); we notice that for a continuously differentiable function f : R n + m → R m , f : ( x , y ) ↦ f ( x , y ) {\displaystyle

an exact differential equation if there exists a continuously differentiable function F, called the potential function, so that ∂ F ∂ x = I {\displaystyle

0)=(1,0)=f(-1,0)} ). Since the differential at a point (for a differentiable function) D f x : T x U → T f ( x ) V {\displaystyle Df_{x}:T_{x}U\to T_{f(x)}V}

of Derrick's Theorem. (Above, f ( s ) {\displaystyle f(s)} is a differentiable function with f ′ ( 0 ) = 0 {\displaystyle f'(0)=0} .) The energy of the

This process is illustrated below. In the case of a continuously differentiable function F, a coordinate descent algorithm can be sketched as: Choose an

functions on convex sets. Given a strictly convex, continuously differentiable function F on a convex set, known as the Bregman generator, the Bregman

_{j\geq 1}(M_{j+1}/M_{j})^{1/j}<\infty } . For any infinitely differentiable function f {\displaystyle f} there are quasi-analytic rings C M {\displaystyle

(where the term dζj is omitted). Suppose that f is a continuously differentiable function on the closure of a domain D in C {\displaystyle \mathbb {C} }

vector field can be realized as the gradient of a continuously differentiable function outside a closed subset of a priori prescribed (small) measure

Poincaré defines a manifold as the level set of a continuously differentiable function between Euclidean spaces that satisfies the nondegeneracy hypothesis

functions, and Markov chains. The Carleman matrix of an infinitely differentiable function f ( x ) {\displaystyle f(x)} is defined as: M [ f ] j k = 1 k

of a Fourier series representing a continuous, almost nowhere-differentiable function, a case not covered by Dirichlet. He also proved the Riemann–Lebesgue

{\displaystyle f\colon \mathbb {R} ^{n}\to \mathbb {R} } is a differentiable function of variables x 1 , … , x n {\displaystyle x_{1},\ldots ,x_{n}}

) {\displaystyle \psi (a+iy)} is an n {\displaystyle n} times differentiable function of y {\displaystyle y} and such that the derivative | d n ψ ( a

the curve as an example of a continuous everywhere yet nowhere differentiable function that was possible to represent geometrically at the time. From

\eta )=g(\xi )} , where g {\displaystyle g} is any real valued differentiable function defined on 1D, the image is invariant to rotations (around the

⊆ R n {\displaystyle \Omega \subseteq \mathbb {R} ^{n}} and a differentiable function f : R n → R n {\displaystyle f:\mathbb {R} ^{n}\rightarrow \mathbb

method is a root-finding algorithm for finding roots of a given differentiable function ⁠ f ( x ) {\displaystyle f(x)} ⁠. The iteration is x n + 1 = x

conformationally relevant dihedral angles, but can in principle be any differentiable function of the cartesian coordinates r {\displaystyle \mathbf {r} } . The

beginning with an initial guess x0. If f is a three times continuously differentiable function and a is a zero of f but not of its derivative, then, in a neighborhood

instance of his more general result: There does not exist a twice-differentiable function u : ℝn → ℝ such that ∂ 2 u ∂ x 1 2 + ⋯ + ∂ 2 u ∂ x n 2 ≥ e u .

instance of his more general result: There does not exist a twice-differentiable function u : ℝn → ℝ such that ∂ 2 u ∂ x 1 2 + ⋯ + ∂ 2 u ∂ x n 2 ≥ e u .

solution of PDE, along with finite element methods. For a n-times differentiable function, by Taylor's theorem the Taylor series expansion is given as f

Differentiable vector-valued functions from Euclidean space – Differentiable function in functional analysis Differentiation in Fréchet spaces Fractal

for i = 1, ..., n − 1, then the n polynomials together define a differentiable function in the interval x0 ≤ x ≤ xn, and for i = 1, ..., n, where If the

differential geometry can be calculated from In(x) if it is a differentiable function. This is the original motivation behind its definition. Vector

literature the Van der Corput lemma. Let f {\displaystyle f} be a real differentiable function in the interval ] a , b ] , {\displaystyle ]a,b],} moreover, inside

the following fact about averages. Given a twice continuously differentiable function f : R n → R {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }

states: if c is a number, and f ( x ) {\displaystyle f(x)} is a differentiable function, then c ⋅ f ( x ) {\displaystyle c\cdot f(x)} is also differentiable

temperature T is an intensive quantity, i.e., a single-valued, differentiable function of three-dimensional space (a scalar field), i.e., that T = T (

SO(3). In a rectangular coordinate system, the gradient of a (differentiable) function f : R 3 → R {\displaystyle f:\mathbb {R} ^{3}\rightarrow \mathbb

right it is expressed in terms of u. If y = f(x) where f is a differentiable function that is invertible, the derivative of the inverse function, if

be taken to find the utility maximising bundle as it is a non differentiable function. Therefore, intuition must be used. The consumer will maximise

_{h\to 0}{\frac {f(x+h)-f(x)}{h}},\qquad \forall x\in [0,1].} Every differentiable function is continuous, so C1([0, 1]) ⊆ C([0, 1]). We claim that ⁠d/dx⁠ :

{\displaystyle x^{*}(q)} may be viewed locally as a continuously differentiable function, and the local response of x ∗ ( q ) {\displaystyle x^{*}(q)} to

These FFFs can provide a second-order approximation to any twice-differentiable function that meets the necessary regulatory conditions, including basic

1/(1-t), which is not defined at t = 1. Nevertheless, if  f  is a differentiable function defined on a compact submanifold of Rn such that the prescribed

known that the modulus (absolute value) of a holomorphic (complex differentiable) function in the interior of a bounded region is bounded by its modulus on

general formula for the surface area of the graph of a continuously differentiable function z = f ( x , y ) , {\displaystyle z=f(x,y),} where ( x , y ) ∈ D

can appear as coefficients in the Taylor series of an infinitely differentiable function defined on the real line, a consequence of Borel's lemma. As a

circles containing an S-like curve represent the application of a differentiable function (like the sigmoid function) to a weighted sum. Peephole convolutional

can be simplified by considering their effect on an arbitrary differentiable function f ( q ) , {\displaystyle f(q),} ( d d q q − q d d q ) f ( q ) =

far as possible from being associated with the level sets of a differentiable function on M (the technical term is "completely nonintegrable tangent hyperplane

worth function, for example, assuming v can be represented as differentiable function of a non-atomic measure on I, μ, v ( c ) = f ( μ ( c ) ) {\displaystyle

satisfactory version of the second fundamental theorem of calculus: each differentiable function is, up to a constant, the integral of its derivative: F ( x ) −

{\displaystyle i} th group/subject, f {\displaystyle f} is a real-valued differentiable function of a group-specific parameter vector ϕ i j {\displaystyle \phi

{\displaystyle k.} If P : U → Y {\displaystyle P:U\to Y} is a continuously differentiable function, then the differential equation x ′ ( t ) = P ( x ( t ) ) , x (

There are two concepts of solution of this problem: a continuously differentiable function u: [0, ∞) → X is called a classical solution of the Cauchy problem

\mathbb {R} ^{mn}} . The argument of J {\displaystyle J} is a differentiable function u : Ω → R m {\displaystyle u:\Omega \to \mathbb {R} ^{m}} , and

f}{\partial t}}\end{aligned}}} where f is any twice continuously differentiable function that depends on position and time. The electromagnetic fields remain

_{0}} is a constant vector, f {\displaystyle f} is any second differentiable function, k ^ {\displaystyle {\hat {\mathbf {k} }}} is a unit vector in

\cdot \mathbf {n} \,dS} Let Velocity U be a twice continuously differentiable function in a region of volume V in space. This function is the stream function

transformation, one starts by considering an arbitrary increasing, twice-differentiable function of r {\displaystyle r} , say G ( r ) {\displaystyle G(r)} . Finding

properties from marginalist theory which assumed utility to be a differentiable function of quantified goods and services. Later work attempted to generalize

the inverse function theorem one can show that a continuously differentiable function f : U → R n {\displaystyle f:U\to \mathbb {R} ^{n}} (where U {\displaystyle

^{2}\geq 0} . Let φ ( x ) {\displaystyle \varphi (x)} be a twice differentiable function, and define the function h ( x ) ≜ φ ( x ) − φ ( μ ) ( x − μ )

{\displaystyle f} is a k {\displaystyle k} -times continuously differentiable function and all derivatives up to the k {\displaystyle k} th one decay

application of differential calculus. The basic way to maximize a differentiable function is to find the stationary points (the points where the derivative

… {\displaystyle s_{x},s_{y},s_{z},\ldots } . Any non-linear differentiable function, f ( a , b ) {\displaystyle f(a,b)} , of two variables, a {\displaystyle

beginning with an initial guess x0. If f is a d + 1 times continuously differentiable function and a is a zero of f but not of its derivative, then, in a neighborhood

be much more difficult. Notice that by adding a suitable known differentiable function to y, whose values at a and b satisfy the desired boundary conditions

properties from marginalist theory which assumed utility to be a differentiable function of quantified goods and services. But it came to be seen that indifference

{\delta G[\rho ](x)}{\delta \rho (y)}}\ .} If G is an ordinary differentiable function (local functional) g, then this reduces to δ F [ g ( ρ ) ] δ ρ

\mathbb {R} ^{2}} , but is not complex differentiable. A real differentiable function is complex differentiable if and only if it satisfies the Cauchy–Riemann

information learned from each event. I(p) is a twice continuously differentiable function of p. Given two independent events, if the first event can yield

\mu )} should converge to a solution of (1). The gradient of a differentiable function h : R n → R {\displaystyle h:\mathbb {R} ^{n}\to \mathbb {R} }

(56) follows that i.e. e {\displaystyle \mathbf {e} } is a smooth differentiable function of the state vector ( r , v ) {\displaystyle (\mathbf {r} ,\mathbf

that giving up this demand means that the wave function is not a differentiable function at the boundary of the box, and thus it can be said that the wave

Let u : M → R {\displaystyle u:M\to \mathbb {R} } be a twice-differentiable function which attains its maximum value C. Suppose that a i j ∂ 2 u ∂ x

dt+\sigma _{t}\,dB_{t}} says that the differential of a twice-differentiable function f(t, x) is given by d f = ( ∂ f ∂ t + μ t ∂ f ∂ x + σ t 2 2 ∂ 2

given as follows: The time evolution of the state is given by a differentiable function from the real numbers R, representing instants of time, to the

∂x⁠ then gives the incompressibility constraint ∇ · v = 0. Any differentiable function may be used for f. The examples that follow use a variety of elementary

{f} ].} Let U be an open set in ℝn, and let φ be a continuously differentiable function from U into the Euclidean space ℝm, where m > n. The mapping φ

} where p is a strictly positive continuously differentiable function and q and r are continuous real-valued functions. For x0 in (a

}{\mathrm {d} x}}f(x)\,,} where i is the imaginary unit and f is a differentiable function of compact support. The multiplication-by-x operator: ( B f ) (

( x , σ ) {\displaystyle f(\mathbf {x} ,\sigma )} be a smooth differentiable function on U ⊂ R 2 × R + {\displaystyle U\subset \mathbb {R} ^{2}\times

continuation shows that the principal nth root is the unique complex differentiable function that extends the usual nth root to the complex plane without the

into an output volume (e.g. holding the class scores) through a differentiable function. A few distinct types of layers are commonly used. These are further

degree k {\displaystyle k} ) is a k + 1 {\displaystyle k+1} times differentiable function f {\displaystyle f} defined by [ f ] ( [ x ] ) := f ( y ) + ∑ i

{f}}(\xi )\right|^{2}.} Suppose f(x) is an absolutely continuous differentiable function, and both f and its derivative f′ are integrable. Then the Fourier

M\}\}_{M=0}=0,} is where O {\displaystyle O} is at least a twice differentiable function on phase space is equivalent to O {\displaystyle O} being a weak

^{+}\rightarrow \mathbb {R} ^{+}} is a strictly increasing and differentiable function such that g ( 0 ) = 0 {\displaystyle g(0)=0} , then we can define

Euler–Maclaurin formula. Assuming that f is a sufficiently often differentiable function the Euler–Maclaurin formula can be written as ∑ k = a b − 1 f (

set up to apply the implicit function theorem. Our continuously differentiable function is the rigidity map ρ : R 2 | V | − 3 → R | E | {\displaystyle

functions. Let 1 < p < ∞ {\displaystyle 1<p<\infty } and define a differentiable function F p : [ 0 , 1 ] → R {\displaystyle F_{p}:[0,1]\rightarrow {\mathbb

under the above redefinition of the potentials for an arbitrary differentiable function F {\displaystyle F} . In order to construct a stochastic Lagrangian

Ω is a bounded open region with smooth boundary ∂Ω and h is a differentiable function on Ω extending to a continuous function on the closure, then, by

the Bochner–Martinelli formula. Suppose that f is a continuously differentiable function on the closure of a domain D on C n {\displaystyle \mathbb {C}

10^{-3}} . The most widely used method for computing a root of any differentiable function f {\displaystyle f} is Newton's method, in which an initial guess

^{n}} be an ⌊ γ ⌋ {\displaystyle \lfloor \gamma \rfloor } -times differentiable function and the ⌊ γ ⌋ {\displaystyle \lfloor \gamma \rfloor } -th derivative

(t_{0})=\mathbf {x} _{0}} , where f {\displaystyle \mathbf {f} } is a differentiable function. Let ( t ) h = { t n : n = 0 , . . , N } {\displaystyle

N_{n})}{\partial \alpha _{i}}}=\{f,H_{i}\},\quad 1\leq i\leq n} for any differentiable function f ( N 1 , … , N n ) {\displaystyle f(N_{1},\dots ,N_{n})} . The

)} . subanalytic subanalytic. subharmonic A twice continuously differentiable function f {\displaystyle f} is said to be subharmonic if Δ f ≥ 0 {\displaystyle